Contractibility for right-angled Artin groups with the standard generating set

Determine whether every right-angled Artin group A_Γ has a contractible Vietoris–Rips complex VR_t(A_Γ) for some finite t when A_Γ is equipped with its standard generating set and standard word metric.

Background

The paper proves that if the defining graph Γ of a right-angled Artin group is triangle-free, then VR_2(A_Γ) is contractible for the standard word metric. This establishes the result for two-dimensional right-angled Artin groups, including groups whose defining graphs have no 3-cliques.

The authors distinguish this result from the general case: although every right-angled Artin group admits some finite generating set yielding contractible Vietoris–Rips complexes, contractibility for the standard generating set remains unresolved in general. The later prediction that VR_t(A_Γ) should be contractible for all t at least the clique number is not included as a separate problem because it is presented without an explicit uncertainty marker.

References

As a remark, all right-angled Artin groups admit finite generating sets with respect to which they have contractible Vietoris--Rips complexes Theorem~4.1 and Lemma~5.20, so the focus here is on the standard generating set, where the contractibility question is open in general.

— Word length, Morse theory, and Vietoris-Rips complexes  (2608.25614 - Hulbert et al., 26 Aug 2026) in Section 1, Introduction, paragraph following Theorem 1.2